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104 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

The probability density function of a continuous random variable can be determined from
the cumulative distribution function by differentiating. Recall that the fundamental theorem of
calculus states that
x
d
f 1u2 du f 1x2
dx

Then, given F(x)

dF1x2
f 1x2
dx

as long as the derivative exists.

EXAMPLE 4-5 The time until a chemical reaction is complete (in milliseconds) is approximated by the
cumulative distribution function

0 x 0
F1x2 e 0.01x
1 e 0 x

Determine the probability density function of X. What proportion of reactions is complete
within 200 milliseconds? Using the result that the probability density function is the deriva-
tive of the F(x), we obtain

0 x 0
f 1x2 e 0.01x
0.01e 0 x
The probability that a reaction completes within 200 milliseconds is

P1X 2002 F12002 1 e 2 0.8647.

EXERCISES FOR SECTION 4-3

4-11. Suppose the cumulative distribution function of the Determine the following:
random variable X is (a) P1X 1.82 (b) P1X 1.52
(c) P1X 22 (d) P1 1 X 12
0 x 0 4-13. Determine the cumulative distribution function for
F1x2 • 0.2x 0 x 5 the distribution in Exercise 4-1.
1 5 x 4-14. Determine the cumulative distribution function for
the distribution in Exercise 4-3.
4-15. Determine the cumulative distribution function for
Determine the following:
the distribution in Exercise 4-4.
(a) P1X 2.82 (b) P1X 1.52
4-16. Determine the cumulative distribution function for
(c) P1X 22 (d) P1X 62
the distribution in Exercise 4-6. Use the cumulative distribu-
4-12. Suppose the cumulative distribution function of the
tion function to determine the probability that a component
random variable X is
lasts more than 3000 hours before failure.
4-17. Determine the cumulative distribution function for
0 x 2 the distribution in Exercise 4-8. Use the cumulative distribu-
F1x2 • 0.25x 0.5 2 x 2 tion function to determine the probability that a length
1 2 x exceeds 75 millimeters.